Let $(M^n, g), (N, h)$ be closed Riemannian manifolds. The Dirichlet energy density of a map $u \in C^2(M,N)$ is defined to be \[ e(u) := (g \otimes u^*h)(\mathrm{d}u, \mathrm{d}u). \] In this expression, $\mathrm{d}u \in \Gamma(T^*M \otimes u^* TN)$ is the pushforward of $u$; it is a section of a vector bundle over $M$, where the value at the point $p$ is the pushforward $T_p M \to T_{u(p)} N$, under the canonical identification of maps $T_p M \to T_{u(p)} N$ with elements of $(T_p M)^* \otimes T_{u(p)} N = (T_p M)^* \otimes (u^* TN)_p$.

The Dirichlet energy of $u$ is the total Dirichlet energy density: \[ E(u) := \frac{1}{2} \int_M e(u) \mathrm{vol}_M. \] In other words, \[ E(u) = \frac{1}{2} |\mathrm{d}u|^2_{L^2} \] is half the squared length of $\mathrm{d}u$ in the $L^2$-norm on sections of $T^* M \otimes u^* TN$.

Consider a family of maps $u_t \in C^2(M,N)$ such that $u_0 = u$. Let $U : \mathbb{R} \times M \to N$ be the map defined by $U(t,x) = u_t(x)$. The family of maps generates a section $\frac{\partial U}{\partial t}$ of the bundle $u^*TN$ over $M$ by \[ \left(\frac{\partial U}{\partial t}\right)_p = \left. \frac{\mathrm{d}}{\mathrm{d}t} u_t(p)\right\rvert_{t=0} \in T_{u(p)} N. \] Write $\partial_t U$ for this section.

The bundle $u^*TN$ has a natural connection $\nabla$, which is the pullback by $u$ of the Levi-Civita connection on $TN$. Thus $\nabla (\partial_t U) \in \Gamma(T^* M \otimes u^* TN)$. Since on ($u^*TN$-valued) $0$-forms the connection agrees with the exterior covariant derivative it induces, one may also write $\mathrm{d} (\partial_t U)$ for this section of $T^* M \otimes u^* TN$, or $\mathrm{d}_{u^* TN} (\partial_t U)$ to emphasize the bundle over with respect to which the exterior derivative is being taken.

For fixed $p \in M$, pullback by the constant map $\psi_p : \mathbb{R} \to M$ defined by $\psi_p(t) = p$ allows one to pull back the bundle $T^* M \otimes u^* TN$ over $M$ to one on $\mathbb{R}$; then the covariant derivative $\nabla_{\frac{\mathrm{d}}{\mathrm{d}t}} \psi_p^* \mathrm{d}u_t$ evaluated at $t=0$ is an element of $\psi_p^*(T^*M \otimes u^*TN)(0) = T_p^*M \otimes T_{u(p)} N$. On the other hand, $\psi_p^* \mathrm{d} u_t = \mathrm{d} u_t(p)$. TODO

The Euler-Lagrange equations for the Dirichlet energy functional are derived from \begin{align*} \left. \frac{\mathrm{d}}{\mathrm{d}t} E(u_t)\right\rvert_{t=0} &= \left(\nabla_{\frac{\mathrm{d}}{\mathrm{d}t}} \mathrm{d}u_t, \mathrm{d}u_0 \right)_{L^2} \\ &= \left(\mathrm{d}_{u^* TN} \frac{\partial U}{\partial t}, \mathrm{d}u \right)_{L^2} \\ &= \left( \frac{\partial U}{\partial t}, \mathrm{d}^*_{u^* TN} (\mathrm{d} u) \right)_{L^2}, \end{align*} and so it is necessary and sufficient for $u$ to be a critical point of the functional $E$ that $\mathrm{d}^*_{u^* TN} (\mathrm{d} u) = 0$, or more compactly that $\mathrm{d}^* (\mathrm{d}u) = 0$.

When the source manifold $M$ is a surface (that is, $n = 2$), then the Dirichlet energy is invariant under conformal change. Suppose $\widetilde{g} = e^{2\phi} g$ is conformally equivalent to the metric $g$ on $M$. We compute the adjoint $\mathrm{d}^*$ of the exterior covariant derivative in the $L^2$ norm induced by the pair of metrics $(g,h)$ as well as the adjoint $\widetilde{\mathrm{d}}^*$ induced by the pair of metrics $(\widetilde{g}, h)$. Let $\sigma \in \Gamma(u^*TN)$ and $\tau \in \Gamma(T^*M \otimes u^*TN)$. Note that the exterior derivative $\mathrm{d} \sigma = \mathrm{d}_{u^*TN} \sigma$ does not depend on the metric on $M$. Using conformal_change_formulae, \begin{align*} (\mathrm{d} \sigma, \tau)_{L^2(g)} &= \int_M (g \otimes u^*h)(\mathrm{d}\sigma, \tau) \mathrm{vol}(g) \\ &= \int_M e^{-2\phi} (g \otimes u^*h)(\mathrm{d}\sigma, \tau) e^{2\phi} \mathrm{vol}(g) \\ &= \int_M (\widetilde{g} \otimes u^*h)(\mathrm{d}\sigma, \tau) \mathrm{vol}(\widetilde{g}) \\ &= (\mathrm{d} \sigma, \tau)_{L^2(\widetilde{g})}. \end{align*} Thus $\mathrm{d}^* = \widetilde{\mathrm{d}}^*$ when $n = 2$. The proof reveals what breaks when $n \neq 2$: in general one is forced to integrate a factor of $e^{(n-2)\phi}$ in order to change the norm from $L^2(g)$ to $L^2(\widetilde{g})$. To be pull this factor out of the integral for all $\sigma, \tau$ requires that $e^{(n-2)\phi}$ is a constant; that is, that either $n = 2$ or $\phi$ is a constant.